## anonymous 5 years ago Does the following series converge or diverge?

1. anonymous

$\sum_{n=2}^{infinity} 1/n \sqrt{\ln(n)}$

2. anonymous

I'm wanting to say it converges to 1, but I'm not sure how to prove that. Real analysis wasn't my strong point.

3. anonymous

do you know what test to use?

4. anonymous

my original thought was ratio, but i dont know how to apply that here

5. anonymous

I'm wanting to say it has something to do with the Cauchy Criterion

6. anonymous

...I have no idea what that means..haha

7. anonymous

I don't think we covered that

8. anonymous

Oh, the ratio test should work.

9. anonymous

really? Let me try that really quickly

10. anonymous

Well, that gave me infinity over infinity...maybe i'm doing it wrong?

11. anonymous

Looking up properties of the natural log right now, give me a sec.

12. anonymous

thanks so much!

13. anonymous

Integral test would be easier since i see ln(n) and 1/n there.

14. anonymous

oh that makes sense!

15. anonymous

Ok all i can think of is that $n / (n+1) \le 1$ and $\sqrt(\ln(n+1)) \le n$ and $\sqrt(\ln(n)) \le n$ thus $\left| n/(n+1) * \sqrt(\ln(n+1)) / \sqrt(\ln(n)) \right| \le \left| 1 * n/n \right| \le 1$

16. anonymous

hmm. Ok, let me think about that for a min :)

17. anonymous

hmm actually that might not work cause the ration needs to be less than 1 for the ratio test to work

18. anonymous

when i worked it doing the integral test i got infinity meaning divergent does that seem right?